Find mathematically (and then plot) the posterior distribution for a binomial likelihood with x = 5 successes out of n = 10 trials using five different beta prior distributions. Does the prior make a large difference in the outcome? If so when? To answer this question complete the following: (a) Find the mathematical formula for the Likelihood Function, using the information above and below. Note the following when doing this problem: • Leave the function B(a, 3) in this form (no need to perform the integration as RStudio can do this); • Find the likelihood: As we have a distinct binomial distribution (n=10, x=5), our like- lihood (L(p|n, z)) will simply be the binomial distribution formula (with n and z sub- stituted in, and p as our random variable). Recall that the term likelihood reflects the fact that this probability' is now a function of the proportion (p), and so is no longer a probability, hence the change of name; • Find the Prior Distribution: For the prior distribution we have a Beta distribution: for a Beta distribution the random variable (X) is between 0 and 1, and thus we can let z = p and write the Beta Distribution in terms of p; • In RStudio: beta(1,2) is the function B(a, 3) as seen in the Beta distribution on the denominator. In this case a = 1, 3 = 2. • Find the Posterior Distribution: Using the formula seen in Module 11 Bayesian Tools
Find mathematically (and then plot) the posterior distribution for a binomial likelihood with x = 5 successes out of n = 10 trials using five different beta prior distributions. Does the prior make a large difference in the outcome? If so when? To answer this question complete the following: (a) Find the mathematical formula for the Likelihood Function, using the information above and below. Note the following when doing this problem: • Leave the function B(a, 3) in this form (no need to perform the integration as RStudio can do this); • Find the likelihood: As we have a distinct binomial distribution (n=10, x=5), our like- lihood (L(p|n, z)) will simply be the binomial distribution formula (with n and z sub- stituted in, and p as our random variable). Recall that the term likelihood reflects the fact that this probability' is now a function of the proportion (p), and so is no longer a probability, hence the change of name; • Find the Prior Distribution: For the prior distribution we have a Beta distribution: for a Beta distribution the random variable (X) is between 0 and 1, and thus we can let z = p and write the Beta Distribution in terms of p; • In RStudio: beta(1,2) is the function B(a, 3) as seen in the Beta distribution on the denominator. In this case a = 1, 3 = 2. • Find the Posterior Distribution: Using the formula seen in Module 11 Bayesian Tools
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
Related questions
Question
Transcribed Image Text:Find mathematically (and then plot) the posterior distribution for a binomial likelihood with x =
5 successes out of n = 10 trials using five different beta prior distributions. Does the prior make
a large difference in the outcome? If so when? To answer this question complete the following:
(a) Find the mathematical formula for the Likelihood Function, using the information above
and below.
Note the following when doing this problem:
• Leave the function B(a, 3) in this form (no need to perform the integration as RStudio
can do this);
• Find the likelihood: As we have a distinct binomial distribution (n=10, x=5), our like-
lihood (L(p|n, x)) will simply be the binomial distribution formula (with n and a sub-
stituted in, and p as our random variable). Recall that the term likelihood reflects the
fact that this probability' is now a function of the proportion (p), and so is no longer a
probability, hence the change of name;
• Find the Prior Distribution: For the prior distribution we have a Beta distribution: for a
Beta distribution the random variable (X) is between 0 and 1, and thus we can let z = p
and write the Beta Distribution in terms of p;
• In RStudio: beta(1,2) is the function B(a, 3) as seen in the Beta distribution on the
denominator. In this case a = 1, 3 = 2.
• Find the Posterior Distribution: Using the formula seen in Module 11 Bayesian Tools
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 4 steps with 2 images
Recommended textbooks for you
MATLAB: An Introduction with Applications
Statistics
ISBN:
9781119256830
Author:
Amos Gilat
Publisher:
John Wiley & Sons Inc
Probability and Statistics for Engineering and th…
Statistics
ISBN:
9781305251809
Author:
Jay L. Devore
Publisher:
Cengage Learning
Statistics for The Behavioral Sciences (MindTap C…
Statistics
ISBN:
9781305504912
Author:
Frederick J Gravetter, Larry B. Wallnau
Publisher:
Cengage Learning
MATLAB: An Introduction with Applications
Statistics
ISBN:
9781119256830
Author:
Amos Gilat
Publisher:
John Wiley & Sons Inc
Probability and Statistics for Engineering and th…
Statistics
ISBN:
9781305251809
Author:
Jay L. Devore
Publisher:
Cengage Learning
Statistics for The Behavioral Sciences (MindTap C…
Statistics
ISBN:
9781305504912
Author:
Frederick J Gravetter, Larry B. Wallnau
Publisher:
Cengage Learning
Elementary Statistics: Picturing the World (7th E…
Statistics
ISBN:
9780134683416
Author:
Ron Larson, Betsy Farber
Publisher:
PEARSON
The Basic Practice of Statistics
Statistics
ISBN:
9781319042578
Author:
David S. Moore, William I. Notz, Michael A. Fligner
Publisher:
W. H. Freeman
Introduction to the Practice of Statistics
Statistics
ISBN:
9781319013387
Author:
David S. Moore, George P. McCabe, Bruce A. Craig
Publisher:
W. H. Freeman